ETD Collection

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Author: search.filters.author.Fareo, Adewunmi Gideon

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    Group invariant solutions for a pre-existing fracture driven by a non-Newtonian fluid in permeable and impermeable rock
    (2013-05-02) Fareo, Adewunmi Gideon
    The aim of the thesis is to derive group invariant, exact, approximate analytical and numerical solutions for a two-dimensional laminar, non-Newtonian pre-existing hydraulic fracture propagating in impermeable and permeable elastic media. The fracture is driven by the injection of an incompressible, viscous non-Newtonian fluid of power law rheology in which the fluid viscosity depends on the magnitude of the shear rate and on the power law index n > 0. By the application of lubrication theory, a nonlinear diffusion equation relating the half-width of the fracture to the fluid pressure is obtained. When the interface is permeable the nonlinear diffusion equation has a leak-off velocity sink term. The half-width of the fracture and the net fluid pressure are linearly related through the PKN approximation. A condition, in the form of a first order partial differential equation for the leak-off velocity, is obtained for the nonlinear diffusion equation to have Lie point symmetries. The general form of the leak-off velocity is derived. Using the Lie point symmetries the problem is reduced to a boundary value problem for a second order ordinary differential equation. The leak-off velocity is further specified by assuming that it is proportional to the fracture half-width. Only fluid injection at the fracture entry is considered. This is the case of practical importance in industry. Two exact analytical solutions are derived. In the first solution there is no fluid injection at the fracture entry while in the second solution the fluid velocity averaged over the width of the fracture is constant along the length of the fracture. For other working conditions at the fracture entry the problem is solved numerically by transforming the boundary value problem to a pair of initial value problems. The numerical solution is matched to the asymptotic solution at the fracture tip. Since the fracture is thin the fluid velocity averaged over the width of the fracture is considered. For the two analytical solutions the ratio of the averaged fluid velocity to the velocity of the fracture tip varies linearly along the fracture. For other working conditions the variation is approximately linear. Using this observation approximate analytical solutions are derived for the fracture half-width. The approximate analytical solutions are compared with the numerical solutions and found to be accurate over a wide range of values of the power-law index n and leak-off parameter β. The conservation laws for the nonlinear diffusion equation are investigated. When there is fluid leak-off conservation laws of two kinds are found which depend in which component of the conserved vector the leak-off term is included. For a Newtonian fluid two conservation laws of each kind are found. For a non-Newtonian fluid the second conservation law does not exist. The behaviour of the solutions for shear thinning, Newtonian and shear thickening fluids are qualitatively similar. The characteristic time depends on the properties of the fluid which gives quantitative differences in the solution for shear thinning, Newtonian and shear thickening fluids.
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    Group invariant solutions for a pre-existing fluid-driven fracture in permeable rock
    (2009-05-22T12:18:35Z) Fareo, Adewunmi Gideon
    The propagation of a two-dimensional fluid-driven pre-existing fluid-filled fracture in permeable rock by the injection of a viscous, incompressible Newtonian fluid is considered. The fluid flow in the fracture is laminar. By the application of lubrication theory, a partial differential equation relating the half-width of the fracture to the fluid pressure and leak-off velocity is obtained. The leak-off velocity is an unspecified function whose form is derived from the similarity solution. The model is closed by the adoption of the PKN formulation in which the fluid pressure is proportional to the fracture half-width. The constant of proportionality depends on the material properties of the rock through its Young modulus and Poisson ratio . The group invariant solutions obtained describe hydraulic fracturing in a permeable rock. Results are also obtained for the case in which the rock is impermeable. Applications in which the rate of fluid injection into the fracture and the pressure at the fracture entry are independent of time are analysed. The limiting solution in which the fracture length and fracture half-width grow exponentially with time is derived. Approximate power law solutions for large values of time for the fracture length and volume are derived. Finally, the case in which the fluid is injected by a pump working at a constant rate is investigated. The results are illustrated by computer generated graphs.